Exploring integral solutions for ternary Diophantine equations : A multi-technique approach

This research paper deals with the problem of finding non-zero integral solutions for a ternary Diophantine equation expressed as pw2 – (2p – 1)wz + pz2 = (k2 + 4p – 1)qr. The investigation explores three distinct cases, each examining the exponent r from 1 to 3. Additionally, the paper outlines methodologies for finding solutions when r is greater than 3. The paper tailors a theorem to this ternary equation, utilizing seven unique techniques to address the problem. A crucial aspect of these methodologies involves a transformative approach where w is replaced with x + y and z is replaced with x – y. Through rigorous analysis and the application of these techniques, the paper provides comprehensive insights into the solutions of the aforementioned Diophantine equation, supported by numerical examples and graphical representations.